Riemann's zeta function and beyond

By S. Gelbart, Stephen D. Miller. 2003

84 citations recorded by the source corpus.

In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations of $L$-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties of L-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of L-functions.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/14efceb3b6518a652f042a29fead3714f9e2aed3

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In Algebra and Number Theory

1. W. Schmid
2. Stephen D. Miller
3. A. Perelli
4. G. Moore
5. B. Pioline
6. Frederik Denef
7. Xiaoqing Li
8. R. Munshi

In Mathematical Physics

1. W. Schmid
2. Stephen D. Miller
3. A. Perelli
4. G. Moore
5. B. Pioline
6. Frederik Denef
7. A. Dabholkar
8. Xiaoqing Li

Citations in the corpus

• Exact and Asymptotic Degeneracies of Small Black Holes (2005, unknown): “[51] for an introduction to this subject.”
• A Survey of the Selberg Class of L-Functions, Part I (2005, background): “The nature of the above L -functions is of courseautomorphic ,a nd we refer to Hecke [7], Iwaniec [9] and Bump [2] for the basic theory of such functions (see also the recent survey by Gelbart-Miller [ 6 ]).”
• Precision Counting of Small Black Holes (2005, background): “ On the other hand, by contrast, the Fourier coefficients an of cusp forms of positive weight k have a lot of “scatter” and can only be described by a probability distribution for an/n(k−1)/2 (see e.g. [87] for an introduction to this subject). As we have remarked above, certain supertraces do in fact”
• Summation formulas, from Poisson and Voronoi to the present (2003, background): “; details can be found in [3, 5 ,22], for example.”
• Cancellation in additively twisted sums on GL(n) (2004, unknown): “We now quickly review their basic analytic properties (see for example [13] and [3], [11], [12], [17], [32]).”